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Convergence of Kac--Moody Eisenstein Series over a function field

Kyu-Hwan Lee, Dongwen Liu, Thomas Oliver

TL;DR

This work addresses the convergence of Eisenstein series on symmetrizable Kac--Moody groups over function fields, extending affine/real results to a general Kac--Moody setting without root-system restrictions. The authors construct an adelic framework, establish convergence of constant terms via the Gindikin--Karpelevich formula, and prove everywhere absolute convergence in the Godement range for the spectral parameter and within the Tits cone for group elements. The key novelty is a representation-theoretic method to generate a positive-measure subset inside the unipotent radical, enabling globalization from constant terms to full series convergence. The results have potential implications for Langlands-type methods over function fields and support arithmetic applications of Kac--Moody Eisenstein series in broader automorphic contexts.

Abstract

We establish everywhere convergence in a natural domain for Eisenstein series on a symmetrizable Kac--Moody group over a function field. Our method is different from that of the affine case which does not directly generalize. In comparison with the analogous result over the real numbers, everywhere convergence is achieved without any additional condition on the root system.

Convergence of Kac--Moody Eisenstein Series over a function field

TL;DR

This work addresses the convergence of Eisenstein series on symmetrizable Kac--Moody groups over function fields, extending affine/real results to a general Kac--Moody setting without root-system restrictions. The authors construct an adelic framework, establish convergence of constant terms via the Gindikin--Karpelevich formula, and prove everywhere absolute convergence in the Godement range for the spectral parameter and within the Tits cone for group elements. The key novelty is a representation-theoretic method to generate a positive-measure subset inside the unipotent radical, enabling globalization from constant terms to full series convergence. The results have potential implications for Langlands-type methods over function fields and support arithmetic applications of Kac--Moody Eisenstein series in broader automorphic contexts.

Abstract

We establish everywhere convergence in a natural domain for Eisenstein series on a symmetrizable Kac--Moody group over a function field. Our method is different from that of the affine case which does not directly generalize. In comparison with the analogous result over the real numbers, everywhere convergence is achieved without any additional condition on the root system.
Paper Structure (4 sections, 12 theorems, 38 equations)

This paper contains 4 sections, 12 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kac--Moody Eisenstein series
  4. Everywhere convergence

Key Result

Theorem 1.1

If $\lambda\in\mathfrak{h}^{\ast}_{\mathbb{C}}$ satisfies $Re(\lambda - \rho) \in \mathcal{C}^*$, then the Kac--Moody Eisenstein series $E_\lambda(g)$ converges absolutely for $g\in U_{\mathbb{A}}H_\mathfrak{C}\mathbb{K}$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2: CG
  • Corollary 2.3
  • Proposition 2.4: G80
  • Proposition 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 12 more